An existence result for the mean-field equation on compact surfaces in a doubly supercritical regime (Q2871079)

Let \(\Sigma\) be a compact orientable surface without boundary, with Riemannian metric \(g\) and volume \(|\Sigma|\). Let \(h_1,h_2:\Sigma\to\mathbb{R}^+\) be smooth functions. The author considers the mean-field equation NEWLINE\[NEWLINE-\Delta_g u = \rho_1 \left( \frac{h_1(x)e^u}{\int_\Sigma h_1(x)e^u dV_g} - \frac{1}{|\Sigma|}\right) - \rho_2 \left(\frac {h_2(x)e^{-u}}{\int_\Sigma h_2(x)e^{-u}dV_g}-\frac{1}{|\Sigma|}\right) NEWLINE\]NEWLINE where \(\rho_1,\rho_2> 0\) are parameters. The integer multiples of \(8\pi\) are critical parameters for the existence of solutions. The main theorem of the paper states that there exists a solution \(u : \Sigma\to\mathbb{R}\) for \(\rho_1, \rho_2\in (8\pi, 16\pi)\).NEWLINENEWLINEThe proof is based on variational methods and on a new version of the Moser-Trudinger inequality for functions \(u\in H^1(\Sigma)\) such that both \(e^u\) and \(e^{-u}\) are concentrated near the same point with the same rate of concentration.